Muriel Livernet

Arc operads and arc algebras

Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identied with open subsets of a combinatorial compactication due to Penner of a space closely related to Riemann’s moduli space. Algebras over these operads are shown to be Batalin{Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy. New operad structures on the circle are classied and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.

Open-Closed Homotopy Algebras and Strong Homotopy Leibniz Pairs Through Koszul Operad Theory

In this paper we study the homology of 2 versions of the swiss-cheese operad. We prove that the zeroth homology of these two versions are Koszul operads and relate this to strong homotopy Lebiniz pairs and OCHA, defined by Kajiura and Stasheff in [13].

Derived A(infinity)-algebras in an operadic context

Derived A1 -algebras were developed recently by Sagave. Their advantage over classical A1 -algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A1 -algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad dAs encoding bidgas, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad A1 as a resolution of the operad As encoding associative algebras. We further show that Sagaves definition of morphisms agrees with the infinity- morphisms of dA1 -algebras arising from operadic machinery. We also study the operadic homology of derived A1 -algebras. 16E45, 18D50; 18G55, 18G10

The non-symmetric operad pre-Lie is free

Operads are a specific tool for encoding type of algebras. For instance there are operads encoding associative algebras, commutative and associative algebras, Lie algebras, pre-Lie algebras, dendriform algebras, Poisson algebras and so on. A usual way of describing a type of algebras is by giving the generating operations and the relations among them. For instance a Lie algebra L is a vector space together with a bilinear product, the bracket (the generating operation) satisfying the relations [x, y] = −[y, x] and [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z ∈ L. The vector space of all operations one can perform on n distinct variables in a Lie algebra is Lie(n), the building block of the symmetric operad Lie. Composition in the operad corresponds to composition of operations. The vector space Lie(n) has a natural action of the symmetric group, so it is a symmetric operad. The case of associative algebras can be considered in two different ways. An associative algebra A is a vector space together with a product satisfying the relation (xy)z = x(yz). The vector space of all operations one can perform on n distinct variables in an associative algebra is As(n), the building block of the symmetric operad As. The vector space As(n) has for basis the symmetric group Sn. But, in view of the relation, one can look also at the vector space of all order-preserving operations one can perform on n distinct ordered variables in an associative algebra: this is a vector space of dimension 1 generated by the only operation x1 · · ·xn. So the non-symmetric operad Ãs describing associative algebras is 1-dimensional for each n: this is the terminal object in the category of non-symmetric operads.

On the spectral sequence of the Swiss-cheese operad

We prove that the homology of the Swiss-cheese operad is a Koszul operad. As a consequence, we obtain that the spectral sequence associated to the stratification of the compactification of points on the upper half plane collapses at the second stage, proving a conjecture by A. Voronov. However, we prove that the operad obtained at the second stage differs from the homology of the Swiss-cheese operad.

Koszul duality of the category of trees and bar constructions for operads

In this paper we study a category of trees TI and prove that it is a Koszul category. Consequences are the interpretation of the reduced bar construction of operads of Ginzburg and Kapranov as the Koszul complex of this category, and the interpretation of operads up to homotopy as a functor from the minimal resolution of TI to the category of graded vector spaces. We compare also three different bar constructions of operads. Two of them have already been compared by Shnider-Von Osdol and Fresse.